Find all $g(x)$ such that $f(g(x))=g(f(x))$

Asked May 03 '16 at 01:39

Active May 03 '16 at 01:39

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For a given function $f(x)$ I want to find all functions $g(x)$ such that $f(g(x))=g(f(x))$.

Two solutions are always $g(x)=x$ and $g(x)=f(x)$, but are there any more?

Initially I wondered this for $f(x)=e^x$, but then also wondered about the general case.

asked May 03 '16 at 01:39

Kwin van der Veen

11,858

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Related: In which cases are $(f \circ g)(x) = (g \circ f)(x)$? and If $f \circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else? – Andrew D. Hwang May 03 '16 at 01:46
This depends on the function obviously. For example, if $f$ is invertible, you can set $g(x)=f^{-1}(x)$. In particular, for $f(x)=e^x$, can take $g(x)=\ln(x)$. – Mustafa May 03 '16 at 02:15
A classic result of Ritt gives a complete answer for polynomials and rational functions. – Bill Dubuque May 03 '16 at 04:31

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